Skip to main content

Theoretical BER vs SNR for binary ASK, FSK, and PSK (with MATLAB Code + Simulator)


Bit Error Rate (BER) Equations

In ASK, noise directly affects the signal amplitude, making it the most vulnerable since the data is carried in amplitude changes. In FSK, data is represented by frequency variations, and because noise typically impacts amplitude more than frequency, FSK is more robust than ASK. In PSK, data is encoded in the signal phase, and BPSK specifically uses 180-degree phase shifts, creating the greatest separation between signal points and therefore achieving the lowest bit error rate (BER) for the same power level. BER formulas for ASK, FSK, and PSK modulation schemes.

ASK

BER = 0.5 × erfc(0.5 × √SNR)

FSK

BER = 0.5 × erfc(√(SNR / 2))

PSK

BER = 0.5 × erfc(√SNR)

erfc / Q-function (Click here)

The Mathematical Foundation

The Bit Error Rate is fundamentally derived from the Conditional Probability of Error, where we integrate the Gaussian Noise PDF over the decision region:

\( P_b = \int_{\mathrm{Threshold}}^{\infty} \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-A)^2}{2\sigma^2}} \, dx = Q\!\left(\frac{A}{\sigma}\right) \)

In digital communications, the argument $x$ inside the $Q(x)$ function represents the ratio of the signal's "safety margin" to the noise magnitude. In an AWGN channel, the noise standard deviation is defined as: $\sigma = \sqrt{N_0 / 2}$

BPSK

Distance to boundary is $\sqrt{E_b}$

$x = \frac{\sqrt{E_b}}{\sqrt{N_0/2}}$
$x = \sqrt{\frac{2E_b}{N_0}}$

BFSK

Distance to boundary is $\frac{\sqrt{E_b}}{\sqrt{2}}$

$x = \frac{\sqrt{E_b}/\sqrt{2}}{\sqrt{N_0/2}}$
$x = \sqrt{\frac{E_b}{N_0}}$

BASK

Distance to boundary is $\frac{\sqrt{E_b}}{2}$

$x = \frac{\sqrt{E_b}/2}{\sqrt{N_0/2}}$
$x = \sqrt{\frac{E_b}{2N_0}}$

Why the 0.5 factor?

In 0.5 * erfc(x), the 0.5 represents the 50% probability of transmitting a '0' vs a '1' in a random bitstream.

Eb/N0 vs SNR

While SNR is the power ratio, Eb/N0 is the energy per bit, allowing us to compare schemes with different bandwidths fairly.

Theoretical BER vs SNR for Amplitude Shift Keying (ASK)

The theoretical Bit Error Rate (BER) for binary ASK depends on how binary bits are mapped to signal amplitudes. For typical cases:

If bits are mapped to 1 and -1, the BER is:

BER = Q(√(2 × SNR))

If bits are mapped to 0 and 1, the BER becomes:

BER = Q(√(SNR / 2))

Where:

  • Q(x) is the Q-function: Q(x) = 0.5 × erfc(x / √2)
  • SNR: Signal-to-Noise Ratio
  • N₀: Noise Power Spectral Density

Understanding the Q-Function and BER for ASK

  • Bit '0' transmits noise only
  • Bit '1' transmits signal (1 + noise)
  • Receiver decision threshold is 0.5

BER is given by:

Pb = Q(0.5 / σ), where σ = √(N₀ / 2)

Using SNR = (0.5)² / N₀, we get:

BER = Q(√(SNR / 2))

ASK BER Formula Derivation

Theoretical BER vs SNR for Frequency Shift Keying (FSK)

For binary FSK, the theoretical BER is:

BER = Q(√(SNR))

BER vs SNR for FSK

The Q-function is defined as:

Q(x) = 0.5 × erfc(x / √2)

BER Formula for BFSK

Similarities Between ASK and FSK

  • Both BERs decrease as SNR increases
  • Both use the Q-function for analytical BER calculation
  • FSK generally performs better under noisy conditions

MATLAB Code for Theoretical BER vs SNR

Binary ASK (BASK)

% The code is written by SalimWireless.Com 

clc;
clear all;
close all;

SNRdB = 0:20; 
SNR = 10.^(SNRdB/10); 

BER_th = (1/2) * erfc(0.5 * sqrt(SNR));

semilogy(SNRdB, BER_th, '-rh', 'linewidth', 2.5);
grid on;
title('Theoretical Bit Error Rate vs. SNR for Binary ASK Modulation');
xlabel('SNR (dB)');
ylabel('BER');
legend('Theoretical');
axis([0 20 1e-5 1]);

Binary FSK (BFSK)

% The code is written by SalimWireless.Com 

clc;
clear;
close all;

SNRdB = 0:1:10;              
SNR = 10.^(SNRdB/10);        

BER_th = (1/2) * erfc(sqrt(SNR / 2));

disp('SNR (dB)    Theoretical BER');
disp([SNRdB', BER_th']);

figure;
semilogy(SNRdB, BER_th, '-kh', 'LineWidth', 2);
xlabel('SNR (dB)');
ylabel('Bit Error Rate (BER)');
title('Theoretical BER vs SNR for BFSK');
grid on;

Theoretical BER vs SNR for BPSK -> (Click Here)

BER vs SNR Simulation



Comparison of ASK, FSK, and PSK Performance

Feature ASK (OOK) BFSK BPSK
Power Efficiency Low Medium High
Bandwidth Efficiency High Low High
Noise Immunity Poor (Sensitive to Amp) Good Excellent
Best Used In Fiber Optics, RFID Caller ID, Paging Deep Space, Satellite

Read More

Key Terms Explained

Eb/N0 (Energy per Bit to Noise Power Density):
The normalized SNR measure, also known as the "SNR per bit." It is the most important metric for comparing different modulation schemes regardless of bandwidth. [Read More] 
Q-Function:
The Q-function represents the tail probability of the standard normal distribution. In BER terms, it tells us the probability that noise will exceed the decision boundary. Read more
Coherent vs. Non-Coherent Detection:
Coherent detection (used in the formulas above) requires the receiver to be in phase-sync with the transmitter. Non-coherent detection is easier to build but results in a higher BER.
Impact of AWGN on BER vs SNR of FSK and PSK →

Further Reading

  1. BER vs SNR for ASK, FSK, and PSK with Online Simulator 
  2. Understanding the Q-function in BASK, BFSK, and BPSK
  3. BER for M-ary PSK and QAM
  4. Constellation Diagrams
📊 MATLAB & ONLINE SIMULATIONS (Web-Based)
🎓 GATE & UGC - NET SOLUTIONS
📡 WIRELESS & SIGNAL PROCESSING


Contact Us

Name

Email *

Message *

Popular Posts

RMS Delay Spread, Excess Delay Spread and Multi-path ...(with MATLAB + Simulator)

📘 Overview of Delay Spread and Multi-path 🧮 Excess Delay spread 🧮 Power delay Profile 🧮 RMS Delay Spread 📚 Further Reading 📂 Other Topics on RMS Delay Spread, Excess Delay ... 🧮 Multipath Components or MPCs 🧮 Online Simulator for Calculating RMS Delay Spread 🧮 Why is there significant multipath in the case of very high frequencies? 🧮 Why RMS Delay Spread is essential for wireless communication? 🧮 Why the Power Delay Profile is essential? 🧮 MATLAB Codes for Calculating Different Types of delay Spreads Delay Spread, Excess Delay Spread, and Multipath (MPCs) The fundamental distinction between wireless and wired connections is that in wireless connections signal reaches at receiver thru multipath signal propagation rather than directed transmission like co-axial cable. Wireless Communication has no set communication path between the transmitter and the receiver. The line...
Read more

Online Simulator for ASK, FSK, and PSK Signal Generation

Interactive Digital Signal Processing (DSP) Tutorial and Simulator for ASK, FSK, and BPSK modulation techniques. Try our new Digital Signal Processing Simulator!   •   Interactive ASK, FSK, and BPSK tools updated for 2025. Start Now Digital Modulation Visualizer: ASK, FSK, & BPSK Simulator Learn and visualize binary modulation techniques (ASK, FSK, BPSK) in real-time with adjustable carrier and sampling parameters. Perfect for DSP students and engineers. 📡 ASK Simulator 📶 FSK Simulator 🎚️ BPSK Simulator 📚 More Topics ASK Modulator FSK Modulator BPSK Modulator More Topics 1. ASK (Amplitude Shift Keying) Simulat...
Read more

Hybrid Beamforming | Page 1

Beamforming Techniques Hybrid Beamforming... Page 1 | Page 2 | Hybrid Beamforming: Hybrid beam formation was developed to address some of the limitations of digital pre-coding approaches. Every antenna element is connected to an RF chain in digital pre-coding (beam forming) method. We also know that each RF chain is in charge of providing a separate data stream between the transmitter and the receiver. We know that a larger number of independent data streams leads to higher data rates. It has a spatial multiplexing feature for MIMO. As a result, we may assume that switching from MIMO to massive MIMO will benefit us more in terms of spatial multiplexing in massive MIMO, where each antenna is coupled to a single RF chain. We'll proceed with a definition of hybrid beam forming. Overview of hybrid beam forming with example: Unlike digital beam forming, more than one antenna element is connected to a single RF chain in hybr...
Read more

Amplitude Shift Keying (ASK) Modulation & Demodulation (with Simulation)

Amplitude Shift Keying (ASK): Signal Analysis and Characterization Theoretical Overview: Amplitude Shift Keying (ASK) represents a primary digital modulation technique wherein information is encoded through discrete variations in the carrier signal's instantaneous amplitude. In a Binary ASK (BASK) framework, the modulation process maps binary data onto two distinct amplitude levels. Specifically, the binary '1' (mark) is conveyed by a sinusoidal carrier with amplitude A c and frequency f c over a bit interval T b , while the binary '0' (space) is represented by a null signal state. This particular signaling method is widely recognized as On-Off Keying (OOK) . It is technically realized by gating a carrier oscillator with a unipolar baseband sequence, effectively performing a product modulation that shifts the baseband spectrum to the carrier frequency. ASK Transmitter Architecture: ...
Read more

MATLAB Code for Rms Delay Spread

RMS delay spread is crucial when you need to know how much the signal is dispersed in time due to multipath propagation, the spread (variance) around the average. In high-data-rate systems like LTE, 5G, or Wi-Fi, even small time dispersions can cause ISI. RMS delay spread is directly related to the amount of ISI in such systems. RMS Delay Spread [↗] Delay Spread Calculator Enter delays (ns) separated by commas: Enter powers (dB) separated by commas: Calculate   The above calculator Converts Power to Linear Scale: It correctly converts the power values from decibels (dB) to a linear scale. Calculates Mean Delay: It accurately computes the mean excess delay, which is the first moment of the power delay profile. Calculates RMS Delay Spread: It correctly calculates the RMS delay spread, defined as the square root of the second central moment of the power delay profile.   MATLAB Code  clc...
Read more

Comparisons among ASK, PSK, and FSK (with MATLAB + Simulator)

Modulation ASK, FSK & PSK Constellation MATLAB Simulink MATLAB Code Comparisons among ASK, PSK, and FSK 📘 Comparisons among ASK, FSK, and PSK 🧮 Online Simulator Bandwidth 🧮 MATLAB Code BER Analysis 📚 Further Reading 📂 View Other Topics on Comparisons among ASK, PSK, and FSK ... 🧮 Comparisons of Noise Sensitivity, Bandwidth, Complexity, etc. 🧮 MATLAB Code for Constellation Diagrams of ASK, FSK, and PSK 🧮 Online Simulator for ASK, FSK, and PSK Generation 🧮 Online Simulator for ASK, FSK, and PSK Constellation 🧮 Some Questions and Answers Comparisons among ASK, PSK, and FSK Comparison among ASK, FSK, and PSK Parameters ASK FSK PSK Variable Characteristics Amplitude ...
Read more

Frequency Shift Keying (FSK) Modulation & Demodulation (with Simulation)

Frequency Shift Keying (FSK) Theoretical Foundations: Frequency Shift Keying (FSK) is a discrete frequency modulation scheme wherein the digital information is encoded via instantaneous shifts in the carrier signal's frequency. The fundamental implementation is Binary FSK (BFSK), which maps binary data onto two distinct, discrete spectral states. A binary '1' (the "mark" state) is represented by a carrier frequency \( f_1 \), while a binary '0' (the "space" state) corresponds to frequency \( f_2 \). Each symbol is sustained for a bit interval denoted by \( T_b \). FSK Transmitter Characterization: The mathematical model for the modulated BFSK output \( s(t) \) is defined as: \[ s(t) = \begin{cases} A_c \cos(2\pi f_1 t), & \text{for } m = 1 \\ A_c \cos(2\pi f_2 t), & \text{for } m = 0 \end{cases} \] ...
Read more

UGC NET Electronic Science Previous Year Question Papers with Solutions

Home / Engineering & Other Exams / UGC NET 2026 PYQ ⬇️ Download Papers and Solutions 📋 Exam Pattern 💡 Preparation Tips ❓ FAQs 📊 Exam Highlights: Electronic Science (88) Feature Details Junior Research Fellowship (JRF) ₹37,000 + HRA per month Eligibility M.Sc/M.Tech in Electronics (55%) Validity of Certificate JRF (3 Years) | Lectureship (Lifetime) 📥 Download UGC NET Electronics PDFs Complete collection of previous year question papers, answer keys and explanations for Subject Code 88. Start Downloading 📂 View All Question Papers June 2025 - Question Paper Download PDF June 2025 - Solved Paper + Explanation ...
Read more